Theorem bj-clel3gALT 37402

Description: Alternate proof of clel3g 3606. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-clel3gALT	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-clel3gALT

Step	Hyp	Ref	Expression
1		elisset 2822	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐵)
2	1	biantrurd 537	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵)))
3		19.41v 1956	. . 3 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵))
4	2, 3	bitr4di 290	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵)))
5		eleq2 2829	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
6	5	bicomd 224	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥))
7	6	pm5.32i 579	. . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))
8	7	exbii 1855	. 2 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))
9	4, 8	bitrdi 288	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815

This theorem is referenced by: (None)